Editing Lambda calculus (section)

== Typed lambda calculus ==
{{Further|Typed lambda calculus}}
A ''typed lambda calculus'' is a typed [[formalism (mathematics)|formalism]] that uses the lambda-symbol (<math>\lambda</math>) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see [[Typed lambda calculus#Kinds of typed lambda calculi|Kinds of typed lambda calculi]]). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and ''untyped lambda calculus'' a special case with only one type.<ref>Types and Programming Languages, p. 273, Benjamin C. Pierce</ref>

Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed [[imperative programming]] languages. Typed lambda calculi play an important role in the design of [[type system]]s for programming languages; here typability usually captures desirable properties of the program, e.g., the program will not cause a memory access violation.

Typed lambda calculi are closely related to [[mathematical logic]] and [[proof theory]] via the [[Curry–Howard isomorphism]] and they can be considered as the [[internal language]] of classes of [[category theory|categories]], e.g., the simply typed lambda calculus is the language of a [[Cartesian closed category]] (CCC).